Perfect matchings in -regular graphs Noga Alon ∗ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 and Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel; Email: noga@math.tau.ac.il. Vo jtech R¨odl † Department of Mathematics and Computer Science, Emory University, Atlanta, USA; Email: rodl@mathcs.emory.edu. Andrzej Ruci´nski ‡ Department of Discrete Mathematics, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna´n, Poland; Email: rucinski@math.amu.edu.pl. Submitted: December 10, 1997; Accepted: February 8, 1998. Abstract Asuper(d, )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V 1 and V 2 , where |V 1 | = |V 2 | = n, in which the minimum degree and the maximum degree are between (d − )n and (d + )n, and for every U ⊂ V 1 ,W ⊂ V 2 with |U|≥n, |W |≥n, | e(U,W ) |U ||W | − e(V 1 ,V 2 ) |V 1 ||V 2 | | <.We prove that for every 1 >d>2>0andn>n 0 (), the number of perfect matchings in any such graph is at least (d − 2) n n! and at most (d +2) n n!. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Br´egman, and the van der Waerden conjecture, proved by Falikman and Egorichev. ∗ Research supported in part by a USA Israeli BSF grant, by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University and by a State of New Jersey grant. † Research supported by Polish-US NSF grant INT-940671 and by NSF grant DMS-9704114. ‡ Research supported by Polish-US NSF grant INT-940671 and by KBN grant 2 P03A 023 09. 0 Mathematics Subject Classification (1991); primary 05C50, 05C70; secondary 05C80 1 the electronic journal of combinatorics 5 (1998), #R13 2 An -regular graph on 2n vertices is a bipartite graph on the classes of vertices V 1 and V 2 ,where|V 1 |=|V 2 |=n,inwhichforeveryU⊂V 1 ,W ⊂ V 2 with |U|≥n, |W |≥n, e(U, W ) |U||W | − e(V 1 ,V 2 ) |V 1 ||V 2 | <, (1) where here e(X, Y ) denotes the number of edges between X and Y .Thequantity e(V 1 ,V 2 ) |V 1 ||V 2 | is called the density of the graph. Such a graph is a super (d, )-regular graph if, in addition, its minimum degree δ and its maximum degree ∆ satisfy (d − )n ≤ δ ≤ ∆ ≤ (d + )n. In this note we prove the following result Theorem 1 Let G be a super (d, )-regular graph on 2n vertices, where d>2and n>n 0 (). Then the number M(G) of perfect matchings of G satisfies (d − 2) n n! ≤ M(G) ≤ (d +2) n n!. Thus, the number of perfect matchings in any super (d, )-regular graph on 2n vertices is close to the expected number of such matchings in a random bipartite graph with edge probability d (which is clearly d n n!). This result is combined with some addi- tional ideas in [7] to derive a new proof of the Blow-Up Lemma of Koml´os, S´ark¨ozy and Szemer´edi. The upper bound in Theorem 1 is true for all bipartite graphs with maximum degree at most (d + )n on at least one side, and is an easy consequence of the Minc conjecture [6] for permanents, proved by Br´egman [2] (c.f. also [1] for a probabilistic description of a proof of Schrijver). Indeed, the Minc conjecture states that the permanent of an n by n matrix A with (0, 1) entries satisfies per(A) ≤ n i=1 r i ! 1/r i , where r i is the sum of the entries of the i-th row of A. To derive the upper bound in Theorem 1 apply this estimate to the matrix A =(a u,v ) u∈V 1 ,v∈V 2 in which a u,v =1if u, v are adjacent and a u,v = 0 otherwise. Here M(G)=per(A). Since the function x! 1/x is increasing, M(G) ≤ (k!) n/k ,wherek=(d+)n, and the upper bound follows by applying the Stirling approximation formula for factorials. It is worth noting that since every -regular graph with density d and 2n vertices contains, in each color class, at most n vertices of degree higher than (d + )n, some version of the above upper bound is also true for any -regular graph of density d. Namely, one can show that for every d>0, if is sufficiently small as a function of d, then for every -regular graph G on 2n vertices with density d we have M (G) < (d +3) n n! the electronic journal of combinatorics 5 (1998), #R13 3 provided n>n 0 (). To prove the lower bound observe that by the van der Waerden conjecture, proved by Falikman [4] and Egorichev [3], the number of perfect matchings in a bipartite k- regular graph with n vertices in each color class is at least (k/n) n n!. Thus it suffices to show that our graph contains a spanning k-regular subgraph (a k-factor), where k = (d − 2)n. This is proved in the next lemma. Lemma 2 Let G beasuper(d, )-regular graph on 2n vertices, d>2. Then G contains a spanning k-factor, where k = (d − 2)n. In the proof of this lemma we will apply the following criterion for containing a k-factor, which can be found e.g. in [5], page 70, Thm. 2.4.2. Theorem 3 Let G be a bipartite graph on 2n vertices in the classes V 1 and V 2 , where |V 1 | = |V 2 | = n. Then G has a k-factor if and only if for all X ⊆ V 1 and Y ⊆ V 2 k|X| + k|Y | + e(V 1 − X, V 2 − Y ) ≥ kn . (2) Proof of Lemma 2. We first assume, to simplify the notation and avoid using floor and ceiling signs when these are not crucial, that (d − 2)n is an integer. By Theorem 3, all we need is to prove inequality (2). If |X|+|Y |≥nthen the left- hand side of (2) is at least nk, and we are done. Assume, thus, that |X| + |Y | <n. Without loss of generality we may and will assume that |V 1 − X|≥|V 2 −Y|.If |V 2 −Y|<n, then, since |X| + |Y | <n, it follows that |X| < |V 2 − Y | <nand thus every vertex of V 2 − Y has at least δ −|X|>(d−2)n=k neighbors in V 1 − X, implying that e(V 1 − X, V 2 − Y ) ≥ (n −|Y|)k, and showing that the left-hand side of(2)isatleastk|X|+k|Y|+k(n−|Y|)≥kn, as needed. Otherwise, |V 1 − X|≥ |V 2 −Y|≥n, and thus, by the -regularity assumption and the obvious fact that e(V 1 ,V 2 )/(|V 1 ||V 2 |) ≥ d−, it follows that e(V 1 −X, V 2 −Y ) > (d−2)(n−|X|)(n−|Y |). Therefore, the left-hand side of (2) is at least k|X| + k|Y | + e(V 1 − X, V 2 − Y ) ≥ (d − 2)(n|X| + n|Y | +(n−|X|)(n −|Y|)) =(d−2)(n 2 + |X||Y |) ≥ (d − 2)n 2 = kn. This completes the proof. Remark: Note that in the last proof the assumption (1) may be relaxed, as we only used the fact that for every U ⊂ V 1 ,W ⊂ V 2 , of cardinality at least n each, e(W,U) |W ||U| ≥ e(V 1 ,V 2 ) |V 1 ||V 2 | − . For the lower bound in Theorem 1 the assumption about the maximum degree of G as well as the assumption that n is sufficiently large as a function of can also be omitted. the electronic journal of combinatorics 5 (1998), #R13 4 References [1] N. Alon and J. Spencer, The Probabilistic Method, Wiley, New York, 1992. [2] L. M. Br´egman, Some properties of nonnegative matrices and their permanents, Soviet Math. Dokl. 14 (1973), 945-949 [Dokl. Akad. Nauk SSSR 211 (1973), 27-30]. [3] G.P. Egorichev, The solution of the van der Waerden problem for permanents, Dokl. Akad. Nauk SSSR 258 (1981), 1041-1044. [4] D. I. Falikman, A proof of van der Waerden’s conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki 29 (1981), 931-938. [5] L. Lov´asz and M. D. Plummer, Matching Theory, Akad´emiai Kiad´o, Budapest, 1986 [6] H. Minc, Nonnegative Matrices, Wiley, 1988 [7] V. R¨odl and A. Ruci´nski, Perfect matchings in -regular graphs and the Blow-up lemma, submitted. . bound follows by applying the Stirling approximation formula for factorials. It is worth noting that since every -regular graph with density d and 2n vertices contains, in each color class, at. Perfect matchings in -regular graphs Noga Alon ∗ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 and Department of Mathematics, Raymond. number of perfect matchings in a bipartite k- regular graph with n vertices in each color class is at least (k/n) n n!. Thus it suffices to show that our graph contains a spanning k-regular subgraph